Arc length

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Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form solutions in some cases.

If the curve is not already a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. The approach is to use an increasingly larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing—possibly indefinitely, but for smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.

For some curves there is a smallest number that is an upper bound on the length of any polygonal approximation. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length .

on the curve . Denote the distance from to by , which is the length of the line segment connecting the two points.

The arc length of is then defined to be

where the supremum is taken over all possible partitions of and is unbounded.

The arc length is either finite or infinite. If then we say that is rectifiable, and is non-rectifiable otherwise. This definition of arc length does not require that is defined by a differentiable function. In fact in general, the notion of differentiability is not defined on a metric space.

Modern methods

Consider a real function such that and (its derivative with respect to ) are continuous on [a,b]. The length of the part of the graph of between and is found by the formula

which is derived from the distance formula approximating the arc length with many small lines. As the number of line segments increases (to infinity by use of the integral) this approximation becomes an exact value.

If a curve is defined parametrically by and , then its arc length between and is

This is more clearly a consequence of the distance formula where instead of a and , we take the limit. A useful mnemonic is

Derivation

In order to approximate the arc length of the curve, it is split into many linear segments. To make the value exact, and not an approximation, infinitely many linear elements are needed. This means that each element is infinitely small. This fact manifests itself later on when an integral is used.

Begin by looking at a representative linear segment (see image) and observe that its length (element of the arc length) will be the differential . We will call the horizontal element of this distance , and the vertical element .

Another way to obtain the integral formula

Suppose that we have a rectifiable curve given by a function , and that we want to approximate the arc length along between two points in that curve. We can construct a series of rectangle triangles whose concatenated hypotenuses "cover" the arch of curve chosen as it's shown in the figure. To make this a "more functional" method we can also demand that the bases of all those triangles were equal to , so that for each one an associated cathetus will exist, depending on the type of curve and on the chosen arch, being then every hypotenuse equal to , as a result of the Pythagorean theorem. This way, an approximation of would be given by the summation of all those unfolded hypotenuses. Because of it we have that;

To continue, let's algebraically operate on the form in which we calculate every hypotenuse to come to a new expression:

Then, our previous result takes the following look:

Now, the smaller these segments are, the better our looked approximation is; they will be as small as we want doing that tends to zero. This way, develops in , and every incremental quotient becomes into a general , that is by definition . Given these changes, our previous approximation turns into a thinner and at this point exact summation; an integration of infinite infinitesimal segments;

Another Proof (Romil Sirohi's)

We know that the formula for a line integral is

If we set the surface to 1, we will get arc length multiplied by 1, or . If and , then , from when to when . If we set these equations into our formula we get

Arbitrary Curves

Then the curve length may be computed over the interval if and only if as spans from to the curve is traced out once and only once.

Historical methods

Ancient

For much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a rectangular approximation for finding the area beneath a curve with his method of exhaustion, few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation.

Integral form

Before the full formal development of the calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre Fermat.

In 1659 van Heuraet published a construction showing that arc lengrves are non-rectifiable, that is, they have infinite length. There are continuous curves for which any arc on the curve (containing more than a single point) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by for and . Sometimes the Hausdorff dimension and Hausdorff measure are usedent]] at had a slope of

so the tangent line would have the equation

Next, he increased by a small amount to , making segment a relatively good approximation for the length of the curve from to . To find the length of the segment , he used the Pythagorean theorem:

which, when solved, yields

In order to approximate the length, Fermat would sum up a sequence of short segments.

Curves with infinite length

As mentioned above, some curves are non-rectifiable, that is, they have infinite length. There are continuous curves for which any arc on the curve (containing more than a single point) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by for and . Sometimes the Hausdorff dimension and Hausdorff measure are used to "measure" the size of infinite length curves.

Generalization to pseudo-Riemannian manifolds

where is the tangent vector of at . The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves.

Finding the measure

The formula the arc measure is:

where:

is the central angle of the arc in degrees

is the radius of the arc

is the diameter of the arc

Recall that is the circumference of the whole circle, so the formula simply reduces this by the ratio of the arc angle to a full angle (360). By transposing the above formula, you solve for the radius, central angle, or arc length if you know any two of them.